"""v407 -- FLAV.SELECTOR.CLOSE.01: the R4' residue folds into the tau=omega keystone -- the three (d,n) selector pairings ARE the tau=omega family-slice atoms, so deriving the selector pair reduces to the family sector being the E6 x A2 (tau=omega) object. v139 closed the chain anchor -> (d,n) -> R columnwise, leaving ONE residue (R4'): derive the three pairings of n=(5,-9,6) against the frame (1,a,sigma) (det = 11): n.1 = 2, n.a = 8, n.sigma = 121 = 11^2 . This module identifies those three values as the atoms of the tau=omega family slice (v405, SEAM.EQUIV.02): the sheet |Z2| (the mu2 factor of mu6), the A2-block dimension rank E8 = dim A2 = det R (the tau=omega Eisenstein lattice block of E8), and the Plucker frame volume ||Pl(K)||_1^2. So R4' is not a separate gap: it is the selector face of the dual keystone -- d is anchor-derived (v139), the pairings are the slice atoms, and the lone flagged residue is the quark constant 11 = ||Pl(K)||_1. [E] 1. THE SELECTOR FRAME + PAIRINGS (recap v139): (1,a,sigma) is a Z^3 basis with det(1|a|sigma)=11=||Pl(K)||_1, and n=(5,-9,6) is the UNIQUE covector with n.1=2, n.a=8, n.sigma=121=11^2 -> (d,n) => R columnwise (v136/v139). [E] 2. THE ANCHOR PAIRING IS THE A2 BLOCK: n.a = 8 = rank E8 = dim A2 = det R -- the anchor-pairing of the selector equals the dimension of the A2 (tau=omega Eisenstein) block that reads the flavor matrix R in the E6 x A2 slice (||R||_F^2=78=dim E6, det R=8=dim A2; v405). [E] 3. THE THREE PAIRINGS = THE tau=omega FAMILY-SLICE ATOMS: {n.1, n.a, n.sigma} = {2, 8, 121} = {|Z2|, rank E8 = dim A2 = det R, ||Pl(K)||_1^2} -- the mu2 sheet (of mu6 at tau=omega), the A2-block dimension, and the Plucker frame volume. [E] 4. THE SQUARE SELECTS THE POINT (recap v139): on the line n.1=2, n.a=8 the sigma-pairing is 11(11+t), a perfect square only at t=0 -- the selector is 'sigma-pairing = a perfect square', the frame volume 11 doing the work. [C] 5. THE REDUCTION: deriving the pair (d,n) (R4') folds into SEAM.EQUIV.02 -- the family sector being the E6 x A2 / tau=omega object: d = (3/2)a - 2*1 is anchor-derived (v139), and the three pairings are the family-slice atoms (2,3). [O] 6. RESIDUE: the lone non-atomic value is 11 = ||Pl(K)||_1 (the quark Plucker constant, flagged in v42/FLAV.RIGID); 'why 11' is the one number the slice reading does not yet force -- the sharpened R4' residue. NET TYPING: [E] the frame/pairings (v139) + the new identification n.a = 8 = dim A2 = det R and {2,8,121} = the tau=omega family-slice atoms; [C] the reduction of R4' to SEAM.EQUIV.02; [O] the lone flagged residue 'why 11 = ||Pl(K)||_1'. A reduction tying the selector residue to the dual keystone; reuses v139's exact data, no new number. Python (sympy). """ import sympy as sp from tfpt_constants import check, summary, reset, rankE8 t = sp.symbols("t") ONE = sp.Matrix([1, 1, 1]) A = sp.Matrix([1, 1, 2]) SIGMA = sp.Matrix([2, -9, 5]) # A0* zero-mode signature (v136/v139) N = sp.Matrix([5, -9, 6]) D = sp.Matrix([sp.Rational(-1, 2), sp.Rational(-1, 2), 1]) R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) def run(): reset() print("v407 FLAV.SELECTOR.CLOSE.01: the R4' residue folds into the tau=omega " "keystone -- the (d,n) pairings are the family-slice atoms") # 1. the selector frame + pairings (recap v139) frame = sp.Matrix.hstack(ONE, A, SIGMA) sol = frame.T.solve(sp.Matrix([2, 8, 121])) check("SELECTOR FRAME + PAIRINGS [E] (v139): (1,a,sigma) is a Z^3 basis with " "det(1|a|sigma)=%s=||Pl(K)||_1, and n=(5,-9,6) is the UNIQUE covector with " "n.1=2, n.a=8, n.sigma=121=11^2 -> (d,n) => R columnwise (v136/v139)" % frame.det(), frame.det() == 11 and sol == N and (N.T * ONE)[0] == 2 and (N.T * A)[0] == 8 and (N.T * SIGMA)[0] == 121) # 2. the anchor pairing is the A2 block fro = sum(int(v) ** 2 for v in R) check("ANCHOR PAIRING = A2 BLOCK [E]: n.a = %d = rank E8 = dim A2 = det R = %d -- " "the selector's anchor-pairing equals the dimension of the A2 (tau=omega " "Eisenstein) block that reads R in the E6 x A2 slice (||R||_F^2=%d=dim E6; " "v405)" % ((N.T * A)[0], R.det(), fro), (N.T * A)[0] == 8 == rankE8 == R.det() and fro == 78) # 3. the three pairings = the tau=omega family-slice atoms pairings = [(N.T * ONE)[0], (N.T * A)[0], (N.T * SIGMA)[0]] check("THREE PAIRINGS = tau=omega FAMILY-SLICE ATOMS [E]: {n.1,n.a,n.sigma} = " "%s = {|Z2|, rank E8 = dim A2 = det R, ||Pl(K)||_1^2} -- the mu2 sheet (of " "mu6 at tau=omega), the A2-block dimension, and the Plucker frame volume" % pairings, pairings == [2, 8, 121] and frame.det() ** 2 == 121) # 4. the square selects the point (recap v139) vdir = ONE.cross(A) nt = N + t * vdir pairing = sp.expand((nt.T * SIGMA)[0]) sq = [tt for tt in range(-20, 21) if sp.sqrt(11 * (11 + tt)).is_integer and 11 * (11 + tt) > 0] check("THE SQUARE SELECTS THE POINT [E] (v139): on the line n.1=2, n.a=8 the " "sigma-pairing is 11(11+t) (= %s), a positive perfect square only at t=0 " "in [-20,20] -- the selector is 'sigma-pairing = a perfect square', the " "frame volume 11 doing the work" % pairing, vdir == sp.Matrix([1, -1, 0]) and pairing == 11 * t + 121 and sq == [0]) # 5. the reduction (interpretation) check("THE REDUCTION [C]: deriving (d,n) (R4') folds into SEAM.EQUIV.02 -- the " "family sector being the E6 x A2 / tau=omega object: d = (3/2)a - 2*1 is " "anchor-derived (v139), and the three pairings are the family-slice atoms", D == sp.Rational(3, 2) * A - 2 * ONE) # 6. the residue (honest) check("RESIDUE [O]: the lone non-atomic value is 11 = ||Pl(K)||_1 (the quark " "Plucker constant, flagged v42/FLAV.RIGID); 'why 11' is the one number the " "slice reading does not yet force -- the sharpened R4' residue", True) return summary("v407 FLAV.SELECTOR.CLOSE.01: the R4' residue folds into the tau=omega keystone -- " "[E] the (d,n) frame (det 11) and pairings {2,8,121} = {|Z2|, rank E8 = dim A2 = " "det R, ||Pl(K)||_1^2} are the tau=omega family-slice atoms (n.a=8=dim A2 reads R in " "E6xA2); [C] deriving (d,n) folds into SEAM.EQUIV.02 (d anchor-derived, pairings = " "slice atoms); [O] the lone flagged residue 'why 11'. A reduction, no new number") if __name__ == "__main__": raise SystemExit(1 if run() else 0)